Approximate quantum compiler (AQC) returns a wrong circuit when the unitary has a determinant −1

[EgrettaThula]

Environment

• Qiskit Terra version: 0.22.3
• Python version: 3.9.5
• Operating system: Windows 10

What is happening?

When the approximate quantum compiler (AQC) is used to synthesis a unitary which has a determinant −1 (e.g., an "odd parity" permutation matrix), a warning message is displayed:

[...]synthesis\aqc\aqc.py:83: RuntimeWarning: invalid value encountered in power
  su_matrix = target_matrix / np.power(target_det, (1 / matrix_dim))

then it continues its work and generates a circuit. The unitary of the generated circuit, however, is not an approximation to the original one.

How can we reproduce the issue?

unitary = np.ndarray(shape=(8, 8), buffer=np.array([
    [0, 1, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 1, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 1, 0, 0, 0],
    [1, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 1, 0, 0],
    [0, 0, 0, 0, 0, 0, 1, 0],
    [0, 0, 0, 0, 0, 0, 0, 1],
], dtype=int), dtype=int)

num_qubits = 3

cnots = make_cnot_network(num_qubits = num_qubits, network_layout = 'spin', connectivity_type = 'full', depth = 0)

optimizer = L_BFGS_B()
aqc = AQC(optimizer)
approximate_circuit = CNOTUnitCircuit(num_qubits=num_qubits, cnots=cnots)
approximating_objective = DefaultCNOTUnitObjective(num_qubits=num_qubits, cnots=cnots)

aqc.compile_unitary(
    target_matrix=unitary,
    approximate_circuit=approximate_circuit,
    approximating_objective=approximating_objective,
)

approx_matrix = Operator(approximate_circuit).data
error = 0.5 * (np.linalg.norm(approx_matrix - unitary, "fro") ** 2)

print(error)
array_to_latex(approx_matrix)

What should happen?

The code must take this case into account

Any suggestions?

The root cause of this wrong behavior is this line of code:

su_matrix = target_matrix / np.power(target_det, (1 / matrix_dim))

As per numpy.power documentation:

Negative values raised to a non-integral value will return nan. To get complex results, cast the input to complex.

The issue can be simply fixed by replacing this line with:

su_matrix = target_matrix / np.power(target_det, (1 / matrix_dim), dtype=complex)

[ajavadia]

Thanks @EgrettaThula for finding this bug, would you like to make a PR to implement your suggestion? You would just have to make the change you suggest, and add a test case.

[EgrettaThula]

I'll work on it.